6 Differential
In the previous chapter Limits of Functions, we built a rigorous foundation for understanding how functions behave near specific points, using the concept of limits to describe continuity and instantaneous change. This foundation naturally leads to the study of derivatives, one of the core ideas in differential calculus. In this chapter (see Figure 6.1), we introduce the derivative from a purely theoretical perspective, focusing on its formulation, interpretation as a limit, and the mathematical rules that allow us to compute derivatives efficiently. No applications are discussed here — our attention is fully on the concepts, definitions, and techniques essential for mastering differential calculus.
We explore:
- the limit-based definition of the derivative,
- alternative forms of the difference quotient,
- the geometric interpretation as the slope of a tangent line,
- rules of differentiation (power rule, product rule, quotient rule, chain rule),
- derivatives of elementary functions,
- implicit and logarithmic differentiation, = higher-order derivatives and their analytical meaning.
This chapter forms the theoretical backbone for later work involving differential equations, optimization, and applied modeling.
6.1 Concept of Derivative
The concept of the derivative describes how a function changes at a specific point. It provides a way to measure the instantaneous rate of change of a function and serves as the foundation of calculus.
According to the Video above, we explore the derivative through three interconnected perspectives: the limit definition \(\lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}\), the geometric meaning of how secant lines approach the tangent line, and the role of the difference quotient in capturing the function’s average rate of change before it becomes instantaneous.
Understanding the derivative begins with understanding how a function changes. Instead of looking at an entire curve at once, we first ask a simpler question: How fast does the function change between two points? To measure this, we compute the difference quotient:
\[ \frac{f(x+h) - f(x)}{h}, \]
which represents the average rate of change over the interval from \(x\) to \(x+h\). Geometrically, this value corresponds to the slope of the secant line connecting the two points on the graph. However, the derivative is not about the average rate of change — it measures the instantaneous rate of change at a single point. To capture that, we let the second point move closer by making \(h\) smaller.
As \(h \to 0\):
- the secant line begins to rotate,
- its slope changes,
- and it gradually approaches the unique line that just touches the curve at one point.
This limiting line is the tangent line, and its slope is defined as the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}. \]
The following animation visually demonstrates this process. As the value of \(h\) decreases, the secant line approaches the tangent line, illustrating how the concept of a derivative emerges from the limit of the difference quotient.
Key Insights
- The secant line represents the average rate of change of the function over a finite interval \(h\).
- As \(h\) becomes smaller, the secant line rotates and approaches the tangent line—illustrating how the derivative is defined as a limit.
- The moving point \((x_0 + h, f(x_0 + h))\) shows how the function behaves near \(x_0\), giving an intuitive view of the idea of “approaching.”
- The tangent line in the graph represents the instantaneous rate of change at \(x_0\), which is the value of the derivative.
6.2 Differentiation Rules
The following rules allow us to differentiate functions efficiently without using the limit definition.
6.2.1 Constant Rule
If \(c\) is a constant: \[ \frac{d}{dx}(c) = 0 \]
6.2.2 Power Rule
For \(f(x) = x^n\): \[ \frac{d}{dx}(x^n) = n x^{n-1} \]
Example ~ Differentiate:
\[ f(x) = 5x^7 - 3x^3 + 2 \]
Solution:
\[
f'(x) = 35x^6 - 9x^2
\]
6.2.3 Constant Multiple Rule
\[ \frac{d}{dx}[c f(x)] = c f'(x) \]
6.2.4 Sum and Difference Rule
\[ \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) \]
6.2.5 Product Rule
If \(y = u(x)v(x)\): \[ y' = u'v + uv' \]
Example ~ Product Rule:
\[ y = (3x^2 + 1)(x^3 - 4) \]
Solution:
Let \(u = 3x^2 + 1\), \(u' = 6x\)
Let \(v = x^3 - 4\), \(v' = 3x^2\)
\[ y' = u'v + uv' \] \[ y' = 6x(x^3 - 4) + (3x^2 + 1)(3x^2) \] \[ y' = 15x^4 + 3x^2 - 24x \]
6.2.6 Quotient Rule
If \(y = \frac{u}{v}\): \[ y' = \frac{u'v - uv'}{v^2} \]
Example ~ Quotient Rule
\[ y = \frac{x^2 + 1}{x - 3} \]
Solution: \[ y' = \frac{2x(x - 3) - (x^2 + 1)}{(x - 3)^2} \]
6.2.7 Chain Rule
If \(y = f(g(x))\): \[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
Example 4 ~ Chain Rule:
\[ f(x) = \sqrt{3x^2 + 5} \]
Rewrite: \[ f(x) = (3x^2 + 5)^{1/2} \]
\[ f'(x) = \frac{1}{2}(3x^2 + 5)^{-1/2} \cdot 6x \]
\[ f'(x) = \frac{3x}{\sqrt{3x^2 + 5}} \]
6.3 Elementary Function Derivatives
6.3.1 Exponential Functions
\[ \frac{d}{dx}(e^x) = e^x \] \[ \frac{d}{dx}(a^x) = a^x \ln(a) \]
6.3.2 Logarithmic Functions
\[ \frac{d}{dx}(\ln x) = \frac{1}{x} \] \[ \frac{d}{dx}(\log_a x) = \frac{1}{x \ln(a)} \]
Example ~Logarithmic Differentiation
\[ y = (x^2 + 1)\sqrt{x}(3x - 5) \]
Take logs: \[ \ln y = \ln(x^2 + 1) + \frac{1}{2}\ln x + \ln(3x - 5) \]
Differentiate: \[ \frac{y'}{y} = \frac{2x}{x^2+1} + \frac{1}{2x} + \frac{3}{3x - 5} \]
Thus: \[ y' = (x^2 + 1)\sqrt{x}(3x - 5)\left( \frac{2x}{x^2+1} + \frac{1}{2x} + \frac{3}{3x - 5} \right) \]
6.3.3 Trigonometric Functions
\[ \frac{d}{dx}(\sin x) = \cos x \] \[ \frac{d}{dx}(\cos x) = -\sin x \] \[ \frac{d}{dx}(\tan x) = \sec^2 x \]
6.3.4 Inverse Trigonometric Functions
\[ \frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}} \] \[ \frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2} \]
6.4 Implicit Differentiation
Implicit differentiation is used when the relationship between \(x\) and \(y\) is not explicitly written as \(y = f(x)\).
Example 1 ~ Implicit Differentiation:
\[ x^2 + y^2 = 25 \]
Differentiate both sides:
\[ 2x + 2y\frac{dy}{dx} = 0 \]
Thus: \[ \frac{dy}{dx} = -\frac{x}{y} \]
Example 2 ~ Implicit Differentiation:
\[ x^3 + y^3 = 9xy \]
Differentiate: \[ 3x^2 + 3y^2 y' = 9(y + xy') \]
Group terms: \[ (3y^2 - 9x)y' = 9y - 3x^2 \]
Thus: \[ y' = \frac{9y - 3x^2}{3y^2 - 9x} \]
6.5 Logarithmic Differentiation
Logarithmic differentiation is especially useful for:
- variable exponents such as \(x^x\)
- products of multiple factors
- complicated quotients
Example:
\[ y = x^x \]
Take the natural logarithm: \[ \ln y = x \ln x \]
Differentiate: \[ \frac{y'}{y} = \ln x + 1 \]
Thus: \[ y' = x^x (\ln x + 1) \]
6.6 Higher-Order Derivatives
Higher-order derivatives are obtained by repeatedly differentiating a function.
- First derivative: \(y'\)
- Second derivative: \[ y'' = \frac{d^2 y}{dx^2} \]
- Third derivative: \[ y''' = \frac{d^3 y}{dx^3} \]
- \(n\)-th derivative: \[ y^{(n)} \]
Example 1 ~ Higher-Order Derivatives:
If \(y = x^4\):
- \(y' = 4x^3\)
- \(y'' = 12x^2\)
- \(y''' = 24x\)
- \(y^{(4)} = 24\)
Example 2 ~ Higher-Order Derivatives:
\[ y = e^{2x} \]
First derivative: \[ y' = 2e^{2x} \]
Second derivative: \[ y'' = 4e^{2x} \]
\(n\)-th derivative: \[ y^{(n)} = 2^n e^{2x} \] ## References {-}